norm2KV.2sided: Normal Suite: Two Sample, Two Sided, Known Variance

Description

Generates the suite of functions related to the two sample normal experiment with a two-sided alternative hypothesis of interest when the variance is known.

Usage

norm2KV.2sided(sigma, prob, mu0, tau0, mu1, tau1, mu2, tau2)

Arguments

sigma

Scalar. The known standard deviation of the population.

prob

Scalar. The prior probability of the null hypothesis. Must be a value between 0 and 1.

mu0

Scalar. The mean of the normal prior density on theta under the null hypothesis. See documentation for dnorm.

tau0

Scalar. The standard deviation of the normal prior density on theta under the null hypothesis. See documentation for dnorm.

mu1

Scalar. The mean of the normal prior density on mean for sample 1 under the alternative hypothesis. See documentation for dnorm.

tau1

Scalar. The standard deviation of the normal prior density on mean for sample 1 under the alternative hypothesis. See documentation for dnorm.

mu2

Scalar. The mean of the normal prior density on mean for sample 2 under the alternative hypothesis. See documentation for dnorm.

tau2

Scalar. The standard deviation of the normal prior density on mean for sample 2 under the alternative hypothesis. See documentation for dnorm.

Value

logm

norm2KV.2sided Returns a list of three vectors: the log marginal density under the null hypothesis (logm0), the log marginal density under the alternative hypothesis (logm1), the log marginal density (logm). Each are evaluated at the observed data provided. The function takes the following usage:

logm(xbar, n, sigma, prob, mu0, tau0, mu1, tau1, mu2, tau2)

xbar: Matrix with 2 columns. Each column represents the sample mean for each of the two samples.
n: Scalar. The sample size.
Remaining parameters are described above for norm2KV.2sided.

logbf

Returns a vector. The value of the log Bayes Factor given the observed data provided and the prior parameters specified. The function has the following usage:

logbf(xbar, n, sigma, prob, mu0, tau0, mu1, tau1, mu2, tau2)

For details on the arguments, see logm above.

prior

Returns a vector. The value of the prior density. The function takes the following usage:

prior(theta, prob, mu0, tau0, mu1, tau1, mu2, tau2)

theta: Vector. The quantiles at which to evaluate the prior.
Remaining paramters are described above for norm2KV.2sided.

post

Returns a vector. The value of the posterior density. The function takes the following usage:

post(theta, xbar, n, sigma, prob, mu0, tau0, mu1, tau1, mu2,
    tau2)

theta: Vector. The quantiles at which to evaluate the posterior.
xbar: Vector of length 2. Each element represents the sample mean for each of the two samples, respectively.
n: Scalar. The sample size.
Remaining paramters are described above for norm2KV.2sided.

ssd.norm2KV.2sided

Sample size calculations for this particular set-up. The function has the following usage:

ssd.norm2KV.2sided(alpha, w, sigma, prob, mu0, tau0, mu1, 
    tau1, mu2, tau2, m = 2500, minn = 2, maxn = 1000, all = FALSE)

See ssd for more details. The suite-specific parameters are described above for norm2KV.2sided.

Details

norm2KV.2sided is used to generate a suite of functions for a two-sample normal experiment with a two-sided alternative hypothesis when the variance is known and the samples are independent. That is, when $$X_j \sim Normal(\theta_j,\sigma^2)$$ $$H_0: \theta_1 = \theta_2 \quad vs. \quad H_1: \theta_1 \neq \theta_2$$ using the following prior on theta[1] and theta[2] $$\pi(\theta) = u*I(\theta_1=\theta_2)Normal(\mu_0,\tau_0^2) + (1-u)*I(\theta_1 \neq \theta_2) Normal(\mu_1,\tau_1^2)Normal(\mu_2,\tau_2^2)$$ where Normal(mu,tau^2) is Normal density with mean mu and variance tau^2 and u is the prior probability of the null hypothesis (prob). The functions that are generated are useful in examining the prior and posterior densities of the parameter theta, as well as constructing the Bayes Factor and determining the sample size via an average error based approach. The arguments of norm2KV.2sided are passed to each of the additional functions upon their creation as default values. That is, if mu0 is set to 1 in the call to norm2KVV.2sided, each of the functions returned will have the defaualt value of 1 for mu0. If an argument is not specified in the call to norm2KV.2sided, then it remains a required parameter in all functions created.

Examples

Run this code

############################################################
# Generate the suite of functions for a two-sample normal
# with a two-sided test. Consider the hypothesis
#      H0: theta[1]==theta[2]  vs.  H1: theta[1]!=theta[2]
#
# with a known variance of 3.

# generate suite
f7 <- norm2KV.2sided(sigma=3,prob=0.5,mu0=0,tau0=1,mu1=2,tau1=1,mu2=2,tau2=1)

# attach suite
attach(f7)

# calculate the Bayes Factor for the following observed data
#   n = 30, xbar[1] = -1, xbar[2] = 1
logbf(xbar=matrix(c(-1,1),nrow=1,ncol=2),n=30)

# perform sample size calculation with TE bound of 0.5 and weight 0.9
#  - due to a need for a Monte Carlo implementation of this procedure, this
#    problem can take significantly longer to solve, compared to other examples.
#    Thus, for this example, a large error bound and weight were chosen to 
#    decrease computation time while illustrating the function.
ssd.norm2KV.2sided(alpha=0.5,w=0.9)

# detain suite
detach(f7)

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